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Determine z-transform of the following $ x(n)=\{1,2,3,4,5,0,7\} $
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Solution:

$ \begin{array}{ll}\\ x(0)=4 & x(1)=5 \quad x(2)=0 \quad x(3)=7 \\\\ x(-1)=3 & x(-2)=2 \quad x(-3)=1\\ \end{array} $

$ \begin{aligned}\\ x(z) & =\sum_{n=-\infty}^{\infty} x(n) z^{-n} \\\\ & =\sum_{n=-3}^3 x(n) z^{-n}\\ \end{aligned} $

$ \begin{aligned}\\ & =x(-3) z^{+3}+x(-z) z^{+2}+x(-1) z^{+1}+x(0) z^0+ \\\\ & x(1) z^{-1}+x(2) z^{-2}+x(3) z^{-3}\\ \end{aligned}\\ $

$ =1 z^{+3}+2 z^{+2}+3 z^{+1}+4+5 z^{-1}+0 z^{-z}+7 z^{-3}\\ $

$ \begin{aligned} & =z^3+2 z^2+3 z+4+\frac{5}{z}+\frac{7}{z^3} \\\\ x(z) & =z^3+2 z^2+3 z+4+5 / z+7 / z^3\\ \end{aligned}\\ $

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